Console beam vibrations

In this paper, we study the initial boundary value problem for the equation of vibrations of a beam, one end of which is free and the other is closed, i.e., for a cantilever beam. The solution of the problem is carried out by the methods of spectral analysis. For the spectral problem, the eigenvalues as the roots of the transcendental equation are found and the corresponding system of eigenfunctions is constructed. It is shown that the constructed system of eigenfunctions has the properties of orthogonality and completeness in the space L2. The uniqueness of the solution of the problem is proved in two ways. The first method is based on the application of the energy integral, and the second-on the completeness of the system of eigenfunctions. The solution of this initial boundary value problem is constructed as the sum of a series in the system of eigenfunctions of the corresponding one-dimensional spectral problem. Estimates of the coefficients of this series and the system of eigenfunctions are found, on the basis of which sufficient conditions are established for the initial functions, the fulfillment of which ensures uniform convergence of the constructed series in the class of regular solutions of the beam vibration equation. Based on the obtained solution of this problem, the stability of its solution depending on the initial data is established.